A frequency‐domain approach to the analysis of stability and bifurcations in nonlinear systems described by differential‐algebraic equations

A general numerical technique is proposed for the assessment of the stability of periodic solutions and the determination of bifurcations for limit cycles in autonomous nonlinear systems represented by ordinary differential equations in the differential‐algebraic form. The method is based on the harmonic balance (HB) technique, and exploits the same Jacobian matrix of the nonlinear system used in the Newton iterative numerical solution of the HB equations for the determination of the periodic steady state. To demonstrate the approach, it is applied to the determination of the bifurcation curves in the parameters' space of Chua's circuit with cubic nonlinearity, and to the study of the dynamics of the limit cycle of a Colpitts oscillator. Copyright © 2007 John Wiley & Sons, Ltd.     

A discrete-time approach to the steady state analysis and optimization of non-linear autonomous circuits

In this paper a method for the steady state analysis and optimization of non-linear autonomous circuits is described. After discretizing the linear part of the circuit, a system of non-linear algebraic equations is obtained. the final formulation is written entirely in the discrete-time domain, making it unnecessary to repeatedly take direct and inverse DFTs during the solution process. Furthermore, it is shown that the resulting formulation may be viewed as a generalization of the harmonic balance equations. an analytic method for computing the exact partial derivatives of the resulting equations with respect to the samples of the variables, the oscillation period and the circuit element values is described, making the proposed approach efficient for both analysis and optimization. Different globally convergent techniques for solving the non-linear system of equations are described, with emphasis on an algorithm based on fast simulated diffusion. Selected application examples are provided to validate the proposed approach.     

Model order reduction of coupled circuit‐device systems

We consider model order reduction of integrated circuits with semiconductor devices. Such circuits are modeled using modified nodal analysis by differential‐algebraic equations coupled with the nonlinear drift‐diffusion equations. A spatial discretization of these equations with a mixed finite element method yields a high dimensional nonlinear system of differential‐algebraic equations. Balancing‐related model reduction is used to reduce the dimension of the decoupled linear network equations, whereas the semidiscretized semiconductor model is reduced using proper orthogonal decomposition. Because the computational complexity of the reduced‐order model through the nonlinearity of the drift‐diffusion equations still depends on the number of variables of the full model, we apply the discrete empirical interpolation method to further reduce the computational complexity. We provide numerical comparisons that demonstrate the performance of the presented model reduction approach. Copyright © 2012 John Wiley & Sons, Ltd.     

First harmonic analysis: a circuit theory point of view

The first harmonic method is usually applied to nonlinear system analysis, being particularly adequate for the study of oscillations. As shown in this paper, oscillators analysis can be performed by using only basic concepts of circuit theory. The nonlinearities present in the oscillator electric circuits are approximated, via the first harmonic method, by resistance or source-controlled equivalent circuit elements, and the resulting linear circuits are entirely analyzed via conventional circuit theory. As a result, the analysis of electric oscillators and the first harmonic method are introduced in a comprehensible electric circuit context     

A frequency‐domain‐based master stability function for synchronization in nonlinear periodic oscillators

An efficient methodology to study conditions for stable in‐phase synchronization in networks of periodic identical nonlinear oscillators is proposed. The problem of investigating synchronization properties on periodic trajectories is reduced to an eigenvalue problem by means of the joint application of master stability function and harmonic balance techniques. The proposed method permits to exploit the periodicity of trajectories, reducing computational time with respect to traditional time‐domain approaches (which were designed to deal with generic attractors) and good accuracy. In addition, such method can easily deal with networks of nonlinear periodic oscillators described by differential‐algebraic equations, and then both static and dynamic coupling could be studied. Copyright © 2011 John Wiley & Sons, Ltd.     

Global and local parametric diagnosis of analog short‐channel CMOS circuits using homotopy‐simplicial algorithm

The paper offers an algorithm for local and global parametric diagnosis in nonlinear analog circuits, including both identification of the faulty parameters and determination their values. The algorithm exploits a nonlinear algebraic type test equations which may possess multiple solutions, corresponding to different sets of the parameters values which meet the test. To find the solutions, the homotopy concept is applied. Since the test equation is not given in explicit analytical form, the simplicial method is used to trace the homotopy path. The proposed approach can be applied to a broad class of analog circuits, including the complementary metal–oxide–semiconductor circuits fabricated in nanometer technology. The developed diagnostic procedure has been implemented in DELPHI, whereas the required by the algorithm repeated circuit analyses are carried out using IsSPICE 4 and both environments have been joined together. For illustration, two numerical examples are given. Copyright © 2013 John Wiley & Sons, Ltd.     

A simulation tool for the analysis and verification of the steady state of circuit designs

Analogue and microwave design requires accurate and reliable simulation tools and methods to meet the design specifications. System properties are often measured in the steady state. Well-suited algorithms for calculating the steady state can be classified into shooting methods, finite difference methods and the harmonic balance (HB) technique. Harmonic balance is a frequency domain method which approaches the problem of finding the steady state by a trigonometric polynomial. Depending on the size of the circuit and the number of Fourier coefficients of the polynomial, the resulting system of non-linear equations can become very large. These non-linear equations are solved by using Newton's method. The sparse linear system arising from Newton's method can be solved by direct, stationary or non-stationary iterative solvers. Iterative methods are normally easy to parallelize or vectorize. In this paper a tool for the simulation of the steady state of electronic circuits is presented. the steady state is calculated using the harmonic balance technique. Non-linear equations are solved by Newton's method and linear equations by preconditioned non-stationary iterative solvers (CGS, Bi-CGSTAB, BiCGSTAB(2), TFQMR). the run time is reduced dramatically, by up to an order of magnitude.     

多谐波源系统的非迭代式谐波潮流分析

为研究多谐波源系统的电能质量问题,提出一种快速的非迭代式谐波潮流分析算法。该算法将基频下的谐波源视为恒功率负荷,根据基频结果计算谐波源的运行参数及模型。联立求解系统的谐波导纳方程和谐波源模型方程,无需迭代即可得到系统中所有节点的各次谐波电压。该算法可考虑谐波源的各次谐波电压和谐波电流之间的相互耦合,也可考虑系统中多个谐波源之间的相互抵消作用。算法以AC/DC整流装置类谐波源为例阐述,可拓展应用于含其他非线性电力电子装置类谐波源的系统中。以存在多个分散式谐波源的实际系统为例,Matlab编程实现算法并与PSCAD时域仿真对比,结果表明:算法准确度高、计算速度快。     

Small-signal harmonic analysis of non-linear circuits

An approximate harmonic analysis method for non-linear circuits is introduced. the method solves the steady state, time domain representation as well as the frequency response of non-linear circuits. It allows simulation of non-ideal switched capacitor circuits composed of any kinds of linear components and non-linear transistors as switches. All switches must have the same period, but they can be opened and closed at any time. the method proposed may also be applied to mixer analysis for the case of strong LO and weak RF signals. Examples are given to demonstrate that the method is efficient and sufficiently fast to be used in circuit design. the simulation results show good agreement with those obtained by harmonic balance and transient analysis.     

A 3D finite element analysis of large‐scale nonlinear dynamic electromagnetic problems by harmonic balancing and domain decomposition

The dual–primal finite element tearing and interconnecting (FETI‐DP) method is applied together with the harmonic balance method and the fixed‐point (FP) method to improve the efficiency of three‐dimensional finite element analysis of large‐scale nonlinear dynamic electromagnetic problems. The FETI‐DP method decomposes the original problem into smaller subdomains problems. Combined with parallel computing techniques, the total computation time can be reduced significantly. To account for nonlinear B‐H characteristics, the FP method is applied together with the polarization formulation. Because the FP method assumes a constant reluctivity, it decouples the systems of different harmonics. The FETI‐DP method can then be applied to speed up the simulation of each harmonic in each FP iteration. Two benchmark problems and a three‐phase inductor are simulated by the proposed method to validate the formulation and demonstrate its performance. Copyright © 2015 John Wiley & Sons, Ltd.     

Envelope analysis of nonlinear electronic circuits based on harmonic balance method

We propose here a Spice‐oriented envelope analysis based on the HB (harmonic balance) method, where Fourier coefficients are assumed to be slowly varying. The Fourier expansions of nonlinear devices are executed by MATLAB in the symbolic forms. In this time, the nonlinearities need to be approximated by the polynomial functions. The determining equation of the HB method is formulated as Sine–Cosine circuit in the form of schematic diagram using ABMs (analog behavior models) of Spice. Each sub‐circuit corresponding to the higher harmonic component is almost the same circuit topology as the original one and has dynamic elements such as capacitors and inductors. The Sine–Cosine circuit can be solved by the transient analysis of Spice. Thus, our method is rather a symbolic approach in the meaning that the HB determining equation is given by the schematic diagram of Spice. Our method can be easily applied to the analysis of middle order of nonlinear communication circuits such as mixers and amplitude modulators and to the analysis of interesting phenomena in the nonlinear oscillations. After many simulation experiments, the results show that our envelope analysis is about 50 times faster than the direct transient analysis. Copyright © 2010 John Wiley & Sons, Ltd.     

汽轮机组的数学机械化建模与算法分析

对于大型汽轮发电机组,描述系统运行状态的微分方程相对复杂且维数很高,为对其进行解析分析,首先应用模态综合法建模得到降阶后的微分方程组,经代数变换后应用数学机械化方法进行解析建模分析。由于参加消元的节点位移变量均是线性变量,故消元时可保留油膜力表达式的非线性成分不变,这样就得到了维数相对较低且等式右端含有非线性油膜力表达式的代数方程组。为实现具有上述特点的代数方程组的求解,提出了既不同于解析法又与经典数值算法不完全相同的微分控制算法思想,据此实现对复杂汽轮发电机组转子系统的建模及对节点位移响应的分析与预测。     

基于Pisarenko谐波分解的间谐波估算方法

随着电力系统的发展以及非线性电力元件的大量应用,电网中的谐波状况更加复杂化,非整数次谐波分量——间谐波和次谐波变得较为突出。文章应用Pisarenko谐波分解(PHD)原理来计算谐波和间谐波的频率和幅值。该方法基于信号自相关估计将信号的特征空间划分成信号子空间与噪声子空间,根据这两个子空间正交的原理构建特征多项式和代数方程组,从而计算出各种谐波分量的频率和幅值。由于该方法假设各频率分量的相位是随机分布的,无法给出谐波的相位信息。文章还结合整数次谐波、间谐波和次谐波估算仿真实验分析了PHD方法的敏感因素。并针对电力系统中谐波分量和基波分量能量相差较大影响估计结果的问题,提出了低通差分预处理的解决方法,取得了较好的估算效果。     

分岔方法及其在电力系统中的应用

本文给出了可用来分析电力系统电压稳定和周期振荡的分岔理论中的有关转折分岔和Hopf分岔的概念、定理,并根据两类描述电力系统方程：微分方程和代数方程,结合降维方法如LS约化方法、中心流形方法介绍了求解分岔点的方法以及分岔理论在电力系统中的应用。     

基于Walsh函数的SHEPWM技术在电流型多电平变流器中的应用

随着高温超导储能技术的发展,电流型变流器的储能效率问题必将得到解决,电流型多电平变流器(CSC)将在电力系统中得到广泛应用。本文以一种新型单相五电平变流器为例,研究了SHEPWM技术在电流型多电平变流器中的应用。利用基于Walsh函数把其对应的Fourier域内的非线性超越方程组转化为线性代数方程组,求解出了开关角关于基波幅值的分段线形方程,从而可实现输出电流的特定谐波抑制和在线控制。     

同步电机突然两相短路问题的一种新处理方法

本文将谐波平衡法和逐次逼近法应用于解决装有阻尼绕组的同步电机的不对称突然短路问题,并导出了线间(即两相)突然短路后电机各绕组的瞬态电流表达式。     

A switching-parameter algorithm for finding multiple solutions of nonlinear resistive circuits

An efficient algorithm for finding multiple solutions of a system of nonlinear algebraic equations is presented. This algorithm consists of solving an associated system of first order nonlinear differential equations whose independent variable may be switched from one variable to another during each integration step. The choice of the forward Euler predictor and Newton-Raphson corrector for integrating the differential equations leads to an extremely efficient method for implementing this switching-parameter algorithm. This approach involves only the recursive solution of an associated system of linear algebraic equations and can be easily programmed. The switching-parameter algorithm can also be used to derive the driving-point or transfer characteristic curve of multivalued resistive nonlinear networks.     

功率MESFET大信号非线性等效电路的谐波平衡分析

利用谐波平衡（HB）分析方法，对微波大功率MESFET管芯进行了输入，输出匹配电路设计。首先讨论了对非线性MESFET等效电路网络进行非线性与线性网络的划分方法，然后，利用所建立的GaAs MESFET非线性等效电路模型，描述了电路匹配设计的方法，步骤及计算,并且将计算结果与测量结果进行比较，结果表明，方法是成功的，具有非常重要的工程意义，为微波功率GaAs MESFET的实用化奠定了技术基础。     

电-气互联综合能源系统多时段暂态能量流仿真

考虑到气网的慢动态特性,研究了电-气互联综合能源系统的多时段暂态能量流仿真,其中电网采用稳态潮流模型,气网采用暂态能量流模型。运用隐式有限差分法将天然气系统暂态能量流方程转化为非线性代数方程组,然后用牛顿法求解以非线性代数方程组描述的综合能源系统的多时段暂态能量流方程。同时,计及了电力系统与天然气系统2个层面的耦合,即燃气轮机与电转气(P2G)技术。最后,基于修改的IEEE 24节点电力系统和比利时20节点天然气系统,计算其多时段暂态能量流。仿真结果表明,在短时间尺度研究中,天然气系统的稳态模型与暂态模型的计算结果存在显著差异。除此之外,还定量评估了P2G对风电消纳的积极影响。     

基于Walsh变换的特定谐波消除脉宽调制技术在动态电压恢复器中的应用

基于Walsh变换的特定消谐PWM(selective harmonic elimination PWM,SHEPWM)方法把其对应的Fourier域内的超越非线性方程转化为线性代数方程,在合适的初始条件下,能得到基波幅值和开关角的分段线性关系,从而可实现在线调压和谐波抑制,特别适用于动态电压恢复器(dynamic voltage restorer,DVR)逆变器的控制。本文介绍了Walsh函数,分析了Walsh域SHE线性方程的建立和求解过程,提出了以质心PWM技术产生初始开关角分布模式的方法,有效提高了计算效率,并且针对SHEPWM脉冲的谐波分布特点,给出了逆变器的滤波器设计原则和方法。在200 kVA 级DVR上的实验结果表明,逆变器在整个电压范围内能有效抑制各次谐波并有良好的动态特性,证实了该方法的可行性和有效性。